Cauchy's Polynomial Root Theorem
\[p(x+=6+2x^2-6x^4+4x^5-3x^6+x^8\]
.We ca define a companion matrix for
\[p(x)\]
as \[A= \left( \begin{array}{cccccccc} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ -6 & 0 & -2 & 0 & 6 & -4 & 3 & 0 \end{array} \right)\]
Notes that
\[det(P- xI)=p(x)\]
There is an identity matrix in the top right, and the bottom row consists of the negative coefficients in ascending powers of
\[x\]
.The largest coefficient of
\[x\]
is 6. Apply Gershgorin's Theorem to find a restriction on the eigenvalues of this matrix.\[r= Max \{ \| 6 \| , \; 1+ \| 0 \| , \; 1+ \| 2 \| , \; \; 1+ \| 0 \| , \; \; 1+ \| -6 \| , \; 1+ \| 4 \| , \; \; 1+ \| -3 \| , \; 1+ \| 0 \| , \; \; 1+ \| 1 \| \}=7\]
All the roots of
\[p(x)\]
lie in a circle of radius 7This is called Cauchy's polynomial root theorem.
